Homoclinic orbits on non-compact Riemannian manifolds for second order Hamiltonian systems
type: Item
instance of: scholarly article
title: Homoclinic orbits on non-compact Riemannian manifolds for second order Hamiltonian systems
zbMATH Open document ID: 0845.58031
author: Fabio Giannoni
author: Louis Jeanjean
author: Kazunaga Tanaka
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1995-12-18T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108353
review text: Let \(M\) be a complete connected finite dimensional Riemannian manifold of class \({\mathcal C}^3\). A given function \(V(t,x) \in {\mathcal C}^2 (\mathbb{R} \times M, \mathbb{R})\) determines the second order Hamiltonian system \[ D_t \dot x (t) + \text{grad}_x V \bigl( t, x(t) \bigr) = 0, \tag{1} \] where \(\dot x(t)\) denotes the derivative of \(x(t)\) with respect to \(t\), \(D_t \dot x(t)\) the covariant derivative of \(\dot x(t)\) and \(\text{grad}_x V(t,x)\) the gradient of \(V(t,x)\) with respect to the variable \(x\). If \(x_0 \in M\) is such point that \(V(t,x_0) = 0\), \(\text{grad}_x V(t,x_0) = 0\) for all \(t \in \mathbb{R}\) and \(x(t)\) is a solution of (1) such that \(x(t) \to x_0\), \(\dot x(t) \to 0\) as \(t \to \pm \infty\), we say that there is a homoclinic orbit emanating from \(x_0\). The authors show the existence of a homoclinic orbit on \(M\) under several complex technical conditions on the function \(V(t,x)\) in terms of Riemannian gradient and Riemannian Hessian of \(V(t,x)\). This interesting paper is well written and organized.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 34C37
zbMATH DE Number: 826763
zbMATH Keywords: Riemannian structure
zbMATH Keywords: variational method
zbMATH Keywords: Hilbert manifold
zbMATH Keywords: homoclinic orbit
reviewed by: A. Klíč
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 826763